Neural network technologies have taken center stage owing to their powerful computing capability for supporting deep learning in artificial intelligence. However, conventional synaptic devices such as SRAM and DRAM are not satisfactory solutions for neural networks. Recently, several types of memristor devices have become popular alternatives because of their outstanding characteristics such as scalability, high performance, and non-volatility. To understand the characteristics of memristors, a comparison among memristors has been made, considering both maturity and performance. Magneto-resistance random access memory, phase-change random access memory, and resistive random access memory among the proposed memristors are good candidates as synaptic devices for weight storage and matrixvector multiplication required in artificial neural networks (ANNs). Moreover, these devices play key roles as synaptic devices in research for bio-plausible spiking neural networks (SNNs) because their distinctive switching properties are well matched for emulating synaptic and neuron functions of biological neural networks. In this paper we review motivation, advantage, technology, and applications of memristor devices for neural networks from practical approaches of ANNs to futuristic research of SNNs, considering the current status of memristor technology.